In this post, I introduce the concept of optional and predictable projections of jointly measurable processes. Optional projections of right-continuous processes and predictable projections of left-continuous processes were constructed in earlier posts, with the respective continuity conditions used to define the projection. These are, however, just special cases of the general theory. For arbitrary measurable processes, the projections cannot be expected to satisfy any such pathwise regularity conditions. Instead, we use the measurability criteria that the projections should be, respectively, optional and predictable.

The projection theorems are a relatively straightforward consequence of optional and predictable section. However, due to the difficulty of proving the section theorems, optional and predictable projection is generally considered to be an advanced or hard part of stochastic calculus. Here, I will make use of the section theorems as stated in an earlier post, but leave the proof of those until after developing the theory of projection.

As usual, we work with respect to a complete filtered probability space, and only consider real-valued processes. Any two processes are considered to be the same if they are equal up to evanescence. The optional projection is then defined (up to evanescence) by the following.

Theorem 1 (Optional Projection) Let X be a measurable process such that is almost surely finite for each stopping time. Then, there exists a unique optional process , referred to as the optional projection of X, satisfying

(1)

almost surely, for each stopping time .

Predictable projection is defined similarly.

Theorem 2 (Predictable Projection) Let X be a measurable process such that is almost surely finite for each predictable stopping time. Then, there exists a unique predictable process , referred to as the predictable projection of X, satisfying

Consider a probability space and a subset S of . The projection is the set of such that there exists a with . We can ask whether there exists a map

such that . From the definition of the projection, values of satisfying this exist for each individual . By invoking the axiom of choice, then, we see that functions with the required property do exist. However, to be of use for probability theory, it is important that should be measurable. Whether or not there are measurable functions with the required properties is a much more difficult problem, and is answered affirmatively by the measurable selection theorem. For the question to have any hope of having a positive answer, we require S to be measurable, so that it lies in the product sigma-algebra , with denoting the Borel sigma-algebra on . Also, less obviously, the underlying probability space should be complete. Throughout this post, will be assumed to be a complete probability space.

It is convenient to extend to the whole of by setting for outside of . Then, is a map to the extended nonnegative reals for which precisely when is in . Next, the graph of , denoted by , is defined to be the set of with . The property that whenever is expressed succinctly by the inclusion . With this notation, the measurable selection theorem is as follows.

Theorem 1 (Measurable Selection) For any , there exists a measurable such that and

(1)

As noted above, if it wasn’t for the measurability requirement then this theorem would just be a simple application of the axiom of choice. Requiring to be measurable, on the other hand, makes the theorem much more difficult to prove. For instance, it would not hold if the underlying probability space was not required to be complete. Note also that, stated as above, measurable selection implies that the projection of S is equal to a measurable set , so the measurable projection theorem is an immediate corollary. I will leave the proof of Theorem 1 for a later post, together with the proofs of the section theorems stated below.

A closely related problem is the following. Given a measurable space and a measurable function, , does there exist a measurable right-inverse on the image of ? This is asking for a measurable function, , from to such that . In the case where is the Borel space , Theorem 1 says that it does exist. If S is the graph then will be the required right-inverse. In fact, as all uncountable Polish spaces are Borel-isomorphic to each other and, hence, to , this result applies whenever is a Polish space together with its Borel sigma-algebra. (more…)

In contrast to optional processes, the class of predictable processes was used extensively in the development of stochastic integration in these notes. They appeared as integrands in stochastic integrals then, later on, as compensators and in the Doob-Meyer decomposition. Since they are also central to the theory of predictable section and projection, I will revisit the basic properties of predictable processes now. In particular, any of the collections of sets and processes in the following theorem can equivalently be used to define the predictable sigma-algebra. As usual, we work with respect to a complete filtered probability space. However, completeness is not actually required for the following result. All processes are assumed to be real valued, or take values in the extended reals .

Theorem 1 The following collections of sets and processes each generate the same sigma-algebra on .

Compare this with the analogous result for sets/processes generating the optional sigma-algebra given in the previous post. The proof of Theorem 1 is given further below. First, recall that the predictable sigma-algebra was previously defined to be generated by the left-continuous adapted processes. However, it can equivalently be defined by any of the collections stated in Theorem 1. To make this clear, I now restate the definition making use if this equivalence.

Definition 2 The predictable sigma-algebra, , is the sigma-algebra on generated by any of the collections of sets/processes in Theorem 1.

In this post we will describe precisely which processes can be realized as the jumps of a local martingale. This leads to very useful decomposition results for processes — see Theorem 10 below, where we give a decomposition of a process X into martingale and predictable components. As I will explore further in future posts, this enables us to construct particularly useful decompositions for local martingales and semimartingales.

Before going any further, we start by defining the class of local martingales which will be used to match prescribed jump processes. The purely discontinuous local martingales are, in a sense, the orthogonal complement to the class of continuous local martingales.

Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.

The class of purely discontinuous local martingales is often denoted as . Clearly, any linear combination of purely discontinuous local martingales is purely discontinuous. I will investigate in more detail later but, in order that we do have plenty of examples of such processes, we show that all FV local martingales are purely discontinuous.

Lemma 2 Every FV local martingale is purely discontinuous.

Proof: If X is an FV local martingale and M is a continuous local martingale then we can compute the quadratic covariation,

Next, an important property of purely discontinuous local martingales is that they are determined uniquely by their jumps. Throughout these notes, I am considering two processes to be equal whenever they are equal up to evanescence.

Lemma 3 Purely discontinuous local martingales are uniquely determined by their initial value and jumps. That is, if X and Y are purely discontinuous local martingales with and , then .

Proof: Setting we have and . So, M is a continuous local martingale and is a local martingale starting from zero. Hence, it is a supermartingale and we have

So almost surely and, by right-continuity, up to evanescence. ⬜

Note that if X is a continuous local martingale, then the constant process has the same initial value and jumps as X. So Lemma 3 has the immediate corollary.

Corollary 4 Any local martingale which is both continuous and purely discontinuous is almost surely constant.

Recalling that the jump process, , of a cadlag adapted process X is thin, we now state the main theorem of this post and describe precisely those processes which occur as the jumps of a local martingale.

Theorem 5 Let H be a thin process. Then, for a local martingale X if and only if

Then, M is then a martingale and A is an integrable process which is also predictable, in the sense that is -measurable for each . The expected value of the variation of A can be computed in terms of X,

In continuous time, the situation is rather more complex, and will require constraints on the process X other than just integrability. We have already discussed the case for submartingales — the Doob-Meyer decomposition. This decomposes a submartingale into a local martingale and a predictable increasing process.

A natural setting for further generalising the Doob-Meyer decomposition is that of quasimartingales. In continuous time, the appropriate class of processes to use for the component A of the decomposition is the predictable FV processes. Decomposition (2) below is the same as that in the previous post on special semimartingales. This is not surprising, as we have already seen that the class of special semimartingales is identical to the class of local quasimartingales. The difference with the current setting is that we can express the expected variation of A in terms of the mean variation of X, and obtain a necessary and sufficient condition for the local martingale component to be a proper martingale.

As was noted in an earlier post, historically, decomposition (2) for quasimartingales played an important part in the development of stochastic calculus and, in particular, in the proof of the Bichteler-Dellacherie theorem. That is not the case in these notes, however, as we have already proven the main results without requiring quasimartingales. As always, any two processes are identified whenever they are equivalent up to evanescence.

The Doob-Meyer decomposition was a very important result, historically, in the development of stochastic calculus. This theorem states that every cadlag submartingale uniquely decomposes as the sum of a local martingale and an increasing predictable process. For one thing, if X is a square-integrable martingale then Jensen’s inequality implies that is a submartingale, so the Doob-Meyer decomposition guarantees the existence of an increasing predictable process such that is a local martingale. The term is called the predictable quadratic variation of X and, by using a version of the Ito isometry, can be used to define stochastic integration with respect to square-integrable martingales. For another, semimartingales were historically defined as sums of local martingales and finite variation processes, so the Doob-Meyer decomposition ensures that all local submartingales are also semimartingales. Going further, the Doob-Meyer decomposition is used as an important ingredient in many proofs of the Bichteler-Dellacherie theorem.

The approach taken in these notes is somewhat different from the historical development, however. We introduced stochastic integration and semimartingales early on, without requiring much prior knowledge of the general theory of stochastic processes. We have also developed the theory of semimartingales, such as proving the Bichteler-Dellacherie theorem, using a stochastic integration based method. So, the Doob-Meyer decomposition does not play such a pivotal role in these notes as in some other approaches to stochastic calculus. In fact, the special semimartingale decomposition already states a form of the Doob-Meyer decomposition in a more general setting. So, the main part of the proof given in this post will be to show that all local submartingales are semimartingales, allowing the decomposition for special semimartingales to be applied.

The Doob-Meyer decomposition is especially easy to understand in discrete time, where it reduces to the much simpler Doob decomposition. If is an integrable discrete-time process adapted to a filtration, then the Doob decomposition expresses X as

(1)

As previously discussed, M is then a martingale and A is an integrable process which is also predictable, in the sense that is -measurable for each . Furthermore, X is a submartingale if and only if or, equivalently, if A is almost surely increasing.

Moving to continuous time, we work with respect to a complete filtered probability space with time index t ranging over the nonnegative real numbers. Then, the continuous-time version of (1) takes A to be a right-continuous and increasing process which is predictable, in the sense that it is measurable with respect to the σ-algebra generated by the class of left-continuous and adapted processes. Often, the Doob-Meyer decomposition is stated under additional assumptions, such as X being of class (D) or satisfying some similar uniform integrability property. To be as general possible, the statement I give here only requires X to be a local submartingale, and furthermore states how the decomposition is affected by various stronger hypotheses that X may satisfy.

Theorem 1 (Doob-Meyer) Any local submartingale X has a unique decomposition

(2)

where M is a local martingale and A is a predictable increasing process starting from zero.

Furthermore,

if X is a proper submartingale, then A is integrable and satisfies

(3)

for all uniformly bounded stopping times .

X is of class (DL) if and only if M is a proper martingale and A is integrable, in which case

(4)

for all uniformly bounded stopping times .

X is of class (D) if and only if M is a uniformly integrable martingale and is integrable. Then, and exist almost surely, and (4) holds for all (not necessarily finite) stopping times .

A counting process, X, is defined to be an adapted stochastic process starting from zero which is piecewise constant and right-continuous with jumps of size 1. That is, letting be the first time at which , then

By the debut theorem, are stopping times. So, X is an increasing integer valued process counting the arrivals of the stopping times . A basic example of a counting process is the Poisson process, for which has a Poisson distribution independently of , for all times , and for which the gaps between the stopping times are independent exponentially distributed random variables. As we will see, although Poisson processes are just one specific example, every quasi-left-continuous counting process can actually be reduced to the case of a Poisson process by a time change. As always, we work with respect to a complete filtered probability space.

As I will show in this post, compensators of quasi-left-continuous counting processes have many parallels with the quadratic variation of continuous local martingales. For example, Lévy’s characterization states that a local martingale X starting from zero is standard Brownian motion if and only if its quadratic variation is . Similarly, as we show below, a counting process is a homogeneous Poisson process of rate if and only if its compensator is . It was also shown previously in these notes that a continuous local martingale X has a finite limit if and only if is finite. Similarly, a counting process X has finite value at infinity if and only if the same is true of its compensator. Another property of a continuous local martingale X is that it is constant over all intervals on which its quadratic variation is constant. Similarly, a counting process X is constant over any interval on which its compensator is constant. Finally, it is known that every continuous local martingale is simply a continuous time change of standard Brownian motion. In the main result of this post (Theorem 5), we show that a similar statement holds for counting processes. That is, every quasi-left-continuous counting process is a continuous time change of a Poisson process of rate 1. (more…)

The previous post introduced the concept of the compensator of a process, which is known to exist for all locally integrablesemimartingales. In this post, I’ll just look at the very special case of compensators of processes consisting of a single jump of unit size.

Definition 1 Let be a stopping time. The compensator of is defined to be the compensator of .

So, the compensator A of is the unique predictable FV process such that and is a local martingale. Compensators of stopping times are sufficiently special that we can give an accurate description of how they behave. For example, if is predictable, then its compensator is just . If, on the other hand, is totally inaccessible and almost surely finite then, as we will see below, its compensator, A, continuously increases to a value which has the exponential distribution.

However, compensators of stopping times are sufficiently general to be able to describe the compensator of any cadlag adapted process X with locally integrable variation. We can break X down into a continuous part plus a sum over its jumps,

(1)

Here, are disjoint stopping times such that the union of their graphs contains all the jump times of X. That they are disjoint just means that whenever , for any . As was shown in an earlier post, not only is such a sequence of the stopping times guaranteed to exist, but each of the times can be chosen to be either predictable or totally inaccessible. As the first term, , on the right hand side of (1) is a continuous FV process, it is by definition equal to its own compensator. So, the compensator of X is equal to plus the sum of the compensators of . The reduces compensators of locally integrable FV processes to those of processes consisting of a single jump at either a predictable or a totally inaccessible time. (more…)

A very common technique when looking at general stochastic processes is to break them down into separate martingale and drift terms. This is easiest to describe in the discrete time situation. So, suppose that is a stochastic process adapted to the discrete-time filtered probability space . If X is integrable, then it is possible to decompose it into the sum of a martingale M and a process A, starting from zero, and such that is -measurable for each . That is, A is a predictable process. The martingale condition on M enforces the identity

So, A is uniquely defined by

(1)

and is referred to as the compensator of X. This is just the predictable term in the Doob decomposition described at the start of the previous post.

In continuous time, where we work with respect to a complete filtered probability space, the situation is much more complicated. There is no simple explicit formula such as (1) for the compensator of a process. Instead, it is defined as follows.

For an arbitrary process, there is no guarantee that a compensator exists. From the previous post, however, we know exactly when it does. The processes for which a compensator exists are precisely the special semimartingales or, equivalently, the locally integrablesemimartingales. Furthermore, if it exists, then the compensator is uniquely defined up to evanescence. Definition 1 is considerably different from equation (1) describing the discrete-time case. However, we will show that, at least for processes with integrable variation, the continuous-time definition does follow from the limit of discrete time compensators calculated along ever finer partitions (see below).

Although we know that compensators exist for all locally integrable semimartingales, the notion is often defined and used specifically for the case of adapted processes with locally integrable variation or, even, just integrable increasing processes. As with all FV processes, these are semimartingales, with stochastic integration for locally bounded integrands coinciding with Lebesgue-Stieltjes integration along the sample paths. As an example, consider a homogeneous Poisson processX with rate . The compensated Poisson process is a martingale. So, X has compensator .

We start by describing the jumps of the compensator, which can be done simply in terms of the jumps of the original process. Recall that the set of jump times of a cadlag process are contained in the graphs of a sequence of stopping times, each of which is either predictable or totally inaccessible. We, therefore, only need to calculate separately for the cases where is a predictable stopping time and when it is totally inaccessible.

For the remainder of this post, it is assumed that the underlying filtered probability space is complete. Whenever we refer to the compensator of a process X, it will be understood that X is a special semimartingale. Also, the jump of a process is defined to be zero at time .

Lemma 2 Let A be the compensator of a process X. Then, for a stopping time ,

For stochastic processes in discrete time, the Doob decomposition uniquely decomposes any integrable process into the sum of a martingale and a predictable process. If is an integrable process adapted to a filtration then we write . Here, M is a martingale, so that , and A is predictable with . By saying that A is predictable, we mean that is measurable for each . It can be seen that this implies that

Then it is possible to write A and M as

(1)

So, the Doob decomposition is unique and, conversely, the processes A and M constructed according to equation (1) can be seen to be respectively, a predictable process starting from zero and a martingale. For many purposes, this allows us to reduce problems concerning processes in discrete time to simpler statements about martingales and separately about predictable processes. In the case where X is a submartingale then things reduce further as, in this case, A will be an increasing process.

The situation is considerably more complicated when looking at processes in continuous time. The extension of the Doob decomposition to continuous time processes, known as the Doob-Meyer decomposition, was an important result historically in the development of stochastic calculus. First, we would usually restrict attention to sufficiently nice modifications of the processes and, in particular, suppose that X is cadlag. When attempting an analogous decomposition to the one above, it is not immediately clear what should be meant by the predictable component. The continuous time predictable processes are defined to be the set of all processes which are measurable with respect to the predictable sigma algebra, which is the sigma algebra generated by the space of processes which are adapted and continuous (or, equivalently, left-continuous). In particular, all continuous and adapted processes are predictable but, due to the existence of continuous martingales such as Brownian motion, this means that decompositions as sums of martingales and predictable processes are not unique. It is therefore necessary to impose further conditions on the term A in the decomposition. It turns out that we obtain unique decompositions if, in addition to being predictable, A is required to be cadlag with locally finite variation (an FV process). The processes which can be decomposed into a local martingale and a predictable FV process are known as special semimartingales. This is precisely the space of locally integrablesemimartingales. As usual, we work with respect to a complete filtered probability space and two stochastic processes are considered to be the same if they are equivalent up to evanescence.

Theorem 1 is a general version of the Doob-Meyer decomposition. However, the name `Doob-Meyer decomposition’ is often used to specifically refer to the important special case where X is a submartingale. Historically, the theorem was first stated and proved for that case, and I will look at the decomposition for submartingales in more detail in a later post. (more…)